Traditionally, results on matrix completion are conditioned on the structure of the so called specification graph of partial matrices. A partial matrix of order \(n\)  is a matrix whose entries in the \(i\) -th row and the \(j\) -th column are determined if and only if \((i,j)\in {\mathcal {I}}\subseteq [1\!:\!n]^2\)  and are undetermined otherwise. A matrix is said to be partial positive semidefinite/ completely positive/ doubly nonnegative if and only if every fully determined principal submatrix is positive semidefinite/ completely positive/ doubly nonnegative. A partial matrix is positive semidefinite/ completely positive/ doubly nonnegative completable if we can specify the undetermined entries so that the fully specified matrix is semidefinite/ completely positive/ doubly nonnegative. The specification graph of partial matrix \({\mathsf {A}}\)  of order \(n\)  is a graph \(G({\mathsf {A}})\)  with vertices \({\mathcal {V}}= \left\lbrace  v_i\colon i\in [1\!:\!n] \right\rbrace \)  and edges \({\mathcal {E}}\)  such that \(\left\lbrace v_i,v_j \right\rbrace \in {\mathcal {E}}\)  if and only if the entry \(({\mathsf {A}})_{ij}\)  is specified. A symmetric matrix with \(G({\mathsf {A}}) = G\)  is called a symmetric matrix realization of \(G\) . The structural aspects of these graphs that are central when it comes to theorems on matrix completion are chordality and the block-clique structure. A graph is chordal if every cycle with length greater 3 has a chord, i.e. an edge that is not part of the cycle that connects two vertices of that cycle. A block of a graph is a subgraph that is connected, has no disconnected subgraph that is obtained by removing just one vertex and its adjacent edges (i.e. a cut vertex) from the subgraph, and is not contained in any other subgraph with these two properties. A block-clique graph is a graph whose blocks are cliques. For a more elementary introduction to these terms see either [1]}, [2]}.
